%% slac-pub-7113: page file slac-pub-7113-0-0-3.tcx.
%% section 3 Impact on the branching ratio [slac-pub-7113-0-0-3 in slac-pub-7113-0-0-3: slac-pub-7113-0-0-3u1]
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\section{\usemenu{slac-pub-7113::context::slac-pub-7113-0-0-3}{Impact on the branching ratio}}\label{section::slac-pub-7113-0-0-3}
To summarize, we have calculated the virtual
corrections to $b \to s \gamma$ coming from the operators
$O_2$, $O_7$ and $O_8$. The contributions from the other
operators in eq. (\docLink{slac-pub-7113-0-0-1.tcx}[heff]{1}) are either small or vanish.
As discussed above, some of the Bremsstrahlung
corrections to the operator $O_7$ have been transferred into
the matrix element $M_7^{mod}$ for $b \to s \gamma$
in order to simplify the following discussion.
While we have neglected
those virtual corrections of the operators $O_3$-$O_6$
which are given by the analogous diagrams as shown in Fig.
\docLink{slac-pub-7113-0-0-2.tcx}[fig:1]{1},
we took into account the non-vanishing
diagrams of $O_5$ and $O_6$
where the gluon connects the external
quark lines and the photon is radiated from the charm quark;
these corrections are automatically taken into account
when using $C_7^{eff}$ instead of $C_7$.
Since the Wilson coefficients of the omitted
operators are about fifty times smaller than that
of the leading one and since we expect that their matrix
elements can be enhanced at most by color
factors, it seems reasonable to neglect them.
We can now easily
write down the amplitude $A(b \to s \g)$
for $b \to s \gamma$
by summing the various contributions derived in the previous
section. We follow closely the treatment of
Buras et al. \cite{6}, where the general structure
of the next-to-leading order result is discussed in detail.
We write
\be
\label{amplitudevirt}
A(b \to s \g) = -\frac{4 G_F \l_t}{\sqrt{2}} \, \hat{D} \,
\bra s \g|O_7(\mu)|b \ket _{tree}
\ee
with
$\hat{D}$
\be
\label{dhat}
\hat{D} = C_7^{eff}(\mu) + \frac{\a_s(\mu)}{4\p} \left(
C_i^{(0)eff}(\mu) \ell_i \log \frac{m_b}{\mu} +
C_i^{(0)eff} r_i
\right) \quad ,
\ee
and where the quantities $\ell_i$ and $r_i$ are given
for $i=2,7,8$ in eqs. (\docLink{slac-pub-7113-0-0-2.tcx}[l2]{11},\docLink{slac-pub-7113-0-0-2.tcx}[rer2ndr]{12}),
(\docLink{slac-pub-7113-0-0-2.tcx}[l7r7]{17}) and (\docLink{slac-pub-7113-0-0-2.tcx}[l8r8]{23}), respectively.
The first term, $C_7^{eff}(\mu)$,
on the rhs of eq. (\docLink{slac-pub-7113-0-0-3.tcx}[dhat]{25}) has to be
taken up to next-to-leading logarithmic precision in order
to get the full next-to-leading logarithmic result, whereas
it is sufficient to use the leading logarithmic values of
the other Wilson coefficients in eq. (\docLink{slac-pub-7113-0-0-3.tcx}[dhat]{25}).
As the next-to-leading coefficient $C_7^{eff}$ is not known
yet, we replace it in the numerical evaluation by its
leading logarithmic value $C_7^{(0)eff}$.
The notation $\bra s \g|O_7(\mu)|b \ket _{tree}$ in
eq. (\docLink{slac-pub-7113-0-0-3.tcx}[amplitudevirt]{24}) indicates that the explicit $m_b$
factor in the operator $O_7$ is the running mass taken
at the scale $\mu$.
Since the relevant scale for a $b$ quark decay is expected to
be $\mu \sim m_b$, we expand the matrix elements of the
operators around
$\mu=m_b$ up to order $O(\a_s)$.
The result is
\be
\label{amplitudevirtuell}
A(b \to s \g) = -\frac{4 G_F \l_t}{\sqrt{2}} \, D \,
\bra s \g|O_7(m_b)|b \ket _{tree}
\ee
with
$D$
\be
\label{d}
D = C_7^{eff}(\mu) + \frac{\a_s(m_b)}{4\p} \left(
C_i^{(0)eff}(\mu) \gamma_{i7}^{(0)eff} \log \frac{m_b}{\mu} +
C_i^{(0)eff} r_i
\right) \quad ,
\ee
where the quantities $\gamma_{i7}^{(0)eff}=\ell_i + 8 \, \delta_{i7}$
are just the entries of the (effective) leading order anomalous
dimension matrix \cite{6}.
As also pointed out in this reference,
the explicit logarithms of the form
$\a_s(m_b) \log(m_b/\mu)$
in eq. (\docLink{slac-pub-7113-0-0-3.tcx}[d]{27})
are cancelled by the $\mu$-dependence of $C_7^{(0)eff}(\mu)$.
\footnote{As we neglect the virtual correction of $O_3$-$O_6$,
there is a small left-over
$\mu$ dependence.}
Therefore the scale dependence is
significantly reduced by including the virtual corrections
calculated in this paper.
The decay width $\G^{virt}$ which follows
from $A(b \to s \g)$ in eq.
(\docLink{slac-pub-7113-0-0-3.tcx}[amplitudevirtuell]{26}) reads
\be
\label{widthvirt}
\G^{virt} = \frac{m_{b,pole}^5 \, G_F^2 \l_t^2 \a_{em}}{32 \p^4}
\, F \, |D|^2 \quad ,
\ee
where in fact we discard the term of $O(\a_s^2)$
in $|D|^2$.
The factor $F$ in eq. (\docLink{slac-pub-7113-0-0-3.tcx}[widthvirt]{28}) is
\be
F = \left( \frac{m_b(\mu=m_b)}{m_{b,pole}} \right)^2 =
1- \frac{8}{3} \, \frac{\a_s(m_b)}{\p} \quad .
\ee
To get the inclusive decay width for $b \to s \g (g)$, also
the Bremsstrahlung corrections (except the part
we have already absorbed) must be added.
The contribution of the operators
$O_2$ and $O_7$ have been calculated before by
Ali and Greub \cite{3},
recently also
the complete set has been
worked out \cite{4,13,14}.
In the present work we neglect the small contribution
of the operators $O_3$ -- $O_6$ in analogy to the virtual corrections,
where only $O_2$, $O_7$ and $O_8$ were considered also.
The branching ratio $\mbox{BR}(b \to s \g (g))$ is then obtained
by deviding the decay width $\G = \G^{virt} + \G^{brems}$
for $b \to s \g (g)$ by the theoretical expression for the
semileptonic width $\G_{sl}$
(including the well-known $O(\a_s)$ radiative
corrections \cite{21}) and by
multiplying with the measured semileptonic branching ratio
$\mbox{BR}_{sl} = (10.4 \pm 0.4)\%$ \cite{22}.
In Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:3]{3} we present the
result for the branching ratio for $b \to s \g (g)$
based on the contributions discussed above.
A rather crucial parameter is the ratio $m_c/m_b$;
it enters both,
$b \to s \g$ mainly through the
virtual corrections of $O_2$ and
the semileptonic decay width through phase space.
To estimate this ratio we
put $m_{b,pole}=4.8 \pm 0.15$ GeV for the $b$ quark pole mass;
{}from the mass difference $m_b - m_c=3.40$ GeV, which
is known quite precisely through the $1/m_Q$
expansion \cite{23,24},
one then gets $m_c/m_b=0.29 \pm 0.02$.
In the curves we have used the central values for $m_{b,pole}$ and
$m_c/m_b$.
For the CKM matrix elements we put $V_{cb}=V_{ts}$ and $V_{tb}=1$.
In Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:3]{3} we have plotted the calculated branching
ratio as a function of the top quark mass $m_t$.
The horizontal dotted
curves show the CLEO limits for the branching ratio
$\mbox{BR}(B \to X_s \g)$ \cite{2}.
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=bild3l.ps,height=2in,angle=-90}
}
\vspace{0.08in}
\caption[]{Branching ratio for $b \to s \g (g)$
based on the formulae in this section. The different
curves are explained in the text.
\label{fig:3}}
\end{figure}
To illustrate the scale dependence of the branching ratio,
we varied the scale $\mu$ between $(m_b/2)$ and $(2m_b)$.
We considered two 'scenarios' which differ by higher order
terms. First, we
put the scale $\mu=m_b$ in the explicit
$\a_s$ factor in eq. (\docLink{slac-pub-7113-0-0-3.tcx}[d]{27}) and in the correction
to the semileptonic decay width, as it was also done
by Buras et al. \cite{6}. The resulting $\mu$ dependence
is shown by dash-dotted lines.
Second, we retain the scale $\mu$ in the
explicit $\a_s$ factors; this leads to the solid curves in
Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:3]{3}. In both cases the upper curve corresponds
to $\mu=m_b/2$ and the lower curve to $\mu=2m_b$.
We mention that the $\mu$-band is larger in the second
scenario and it is therefore safer to use this band to illustrate
the remaining scale uncertainties.
In Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:4]{4} we show for comparison
the leading logarithmic
result for the branching ratio for $b \to s \g$,
based on the tree-level matrix element of the operator $O_7$
and using the tree-level formula for the semileptonic decay
width. Varying the scale $\mu$ in the same range as above, leads
to the dash-dotted curves in Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:4]{4}.
We have also plotted
the result as is was available
before the inclusion of the virtual corrections of $O_2$ and
$O_8$ (but with Bremsstrahlung and virtual corrections to $O_7$
included).
This is reproduced by putting
$\ell_2=r_2=\ell_8=r_8=0$ in our formulae.
As noticed in the literature \cite{3,14},
the $\mu$ dependence in this case (solid lines) is even larger
than in the leading
logarithmic result.
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=bild4l.ps,height=2in,angle=-90}
}
\vspace{0.08in}
\caption[]{Branching ratio for $b \to s \g(g)$.
The leading logarithmic result is shown by the dash-dotted curve;
The solid line shows the situation before the virtual contributions of
$O_2$ and $O_8$. See text.
\label{fig:4}}
\end{figure}
{}From the results in Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:4]{4}
it was relatively easy to read off
a reasonable
prediction for the branching ratio within a large error
which was essentially determined by the $\mu$ dependence.
In the improved calculation (Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:3]{3}) the $\mu$
dependence is significantly reduced, because all the logarithms
of the form $\a_s(m_b) \log(m_b/\mu)$ are cancelled as discussed
above. In the present situation it is, however,
premature to extract a prediction
for the branching ratio from Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:3]{3}.
This only will be possible
when also $C_7^{eff}$ is known up to next-to-leading
logarithmic precision. But this result will, essentially,
shift the narrow bands of Fig. \docLink{slac-pub-7113-0-0-3.tcx}[fig:3]{3}, without
broadening them significantly. Thus, a very precise
prediction will become possible and renewed experimental
efforts will be required. It is, however,
rewarding to see
that the next-to-leading result will lead to
a strongly improved
determination of the standard model parameters
or to better limits to new physics.
\vspace*{2cm}
{\bf Acknowledgements}
Discussions with A. Ali, S. Brodsky, M. Lautenbacher, M. Peskin
and L. Reina are thankfully acknowledged.
We are particularly indebted to M. Misiak
for many useful comments; especially
his remarks concerning the
renormalization scale dependence were extremely useful.
One of us (C.G.) would like to thank the
Institute for Theoretical Physics in Z\"urich
for the kind hospitality.
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